Question

If $x^2+5y^2+z^2=2y(2x+z)$ then which of the following statement (s) is/are necessarily true?
(i) $x=2y$ (ii) $x=2z$ (iii) $2x=z$

• A. Only (i)
• B. Only (ii) and (iii)
• C. Only (i) and (ii)
• D. Only (ii)

Answer 1

The correct option is C Only (i) and (ii)

$x^2+5y^2+z^2=2y(2x+z)$

$\Rightarrow [x^2+4y^2-2(2y\left)x\right]+(y^2+z^2-yz)=0$

$\Rightarrow (x-2y{)}^2+(y-z{)}^2=0$

As $(x-2y{)}^2\ge 0$ & $(y-z{)}^2\ge 0$

$\Rightarrow (x-2y{)}^2=0$ & $(y-z{)}^2=0$

$\Rightarrow x=2y$ & $y=z\Rightarrow x=2z$